Prove that $f_n$ = $\frac{x}{1+n^2x^2}$ uniformly converges.

Question

Prove that $f_n$ = $\frac{x}{1+n^2x^2}$ uniformly converges.

I know that the limit function would be $f(x)$ = 0 but I'm getting stuck on how to prove that with the definition of uniform convergence.

Answer 1

HINT:

From the AM-GM inequality, $1+n^2x^2\ge 2n|x|$. Thus,

$$\left|\frac{x}{1+n^2x^2}\right|\le \frac1{2n}$$

Answer 2

Hint: Try finding the maximum of $f_n(x)$, and using the definition of uniform convergence. Be sure to consider which interval you're working on.